Finite Group Databasebeta

← #13565

Subgroups — #13565

Order 256.

248 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#455C211yesmaximal, characteristic
2C2×#355C211yesmaximal, characteristic, complemented
3#1533C211yesmaximal, characteristic, complemented
4#1403C214yesmaximal, complemented
5C2×H(5,𝔽2)C411yescharacteristic
6C2×#137—21no—
7C2×#124C2211yescharacteristic
8#455—21no—
9#455—21no—
10#455C412yes—
11#455C2212yes—
12#432—21no—
13C2×#104—21no—
14C2×#101—21no—
15C2×#100C411yescharacteristic
16#355—21no—
17#355C2212yescomplemented
18#329—21no—
19#329C2212yescomplemented
20H(5,𝔽2)—21no—
21H(5,𝔽2)—41no—
22H(5,𝔽2)—42no—
23H(5,𝔽2)C2×C412yes—
24C2×#41—21no—
25C2×#41—41no—
26C22×Q8D411yescharacteristic
27C22×D4—41no—
28C22×D4—41no—
29C22×D4C2×C411yescharacteristic
30#137—22no—
31#137—22no—
32#137—42no—
33C2×Q16—41no—
34C2×SD16—41no—
35C2×M16—21no—
36C2×M16—41no—
37#124—41no—
38#124C2×C411yescharacteristic
39#124D411yescharacteristic
40#124D411yescharacteristic
41#124C2311yescharacteristic
42C2×C22⋊C4—41no—
43C2×C42D411yescharacteristic
44C4⋊C8—41no—
45#104—22no—
46#104—22no—
47#101—22no—
48#101—41no—
49#100—21no—
50#100C2×C412yes—
51C22⋊C8—41no—
52#97—22no—
53C24—21no—
54C24—41no—
55#41—22no—
56#41—22no—
57#41—41no—
58#41—42no—
59#41—44no—
60#41—41no—
61#41—42no—
62C2×Q8—21no—
63C2×Q8—21no—
64C2×Q8—41no—
65C2×Q8—41no—
66C2×Q8C22⋊C411yescharacteristic
67C2×Q8C2×D411yescharacteristic
68C2×D4—21no—
69C2×D4—21no—
70C2×D4—41no—
71C2×D4—41no—
72C2×D4—42no—
73C2×D4—42no—
74C2×D4—42no—
75C2×D4—42no—
76C2×D4—44no—
77C2×D4—42no—
78C2×D4—41no—
79C2×D4—42no—
80C2×D4—42no—
81C2×D4C22⋊C411yescharacteristic
82C2×D4C22×C411yescharacteristic
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—41no—
86C22×C4—41no—
87C22×C4C22⋊C411yescharacteristic
88Q16—42no—
89Q16—42no—
90SD16—42no—
91SD16—42no—
92M16—22no—
93M16—41no—
94M16—42no—
95M16—81no—
96C2×C8—41no—
97C2×C8—42no—
98C2×C8—81no—
99C22⋊C4—41no—
100C22⋊C4—41no—
101C22⋊C4—41no—
102C22⋊C4—41no—
103C42—21no—
104C42C22⋊C411yescharacteristic
105C42C2×D411yescharacteristic
106C23—21no—
107C23—21no—
108C23—42no—
109C23—41no—
110C23—41no—
111C23—42no—
112C23—41no—
113C23—82no—
114C23—162no—
115C23#9911yescharacteristic
116Q8—22no—
117Q8—41no—
118Q8—41no—
119Q8—41no—
120Q8—81no—
121D4—22no—
122D4—41no—
123D4—41no—
124D4—41no—
125D4—42no—
126D4—44no—
127D4—44no—
128D4—42no—
129D4—42no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—42no—
143C2×C4—81no—
144C2×C4#9911yescharacteristic
145C2×C4C2×C22⋊C411yescharacteristic
146C8—42no—
147C8—82no—
148C8—82no—
149C22—41no—
150C22—42no—
151C22—41no—
152C22—41no—
153C22—81no—
154C22—81no—
155C22—82no—
156C22—82no—
157C22—161no—
158C22—162no—
159C22—162no—
160C22AΣL(1,𝔽16)11yescharacteristic
161C22AΣL(1,𝔽16)11yescharacteristic
162C22C2×#9911yescharacteristic
163C4—21no—
164C4—21no—
165C4—41no—
166C4—41no—
167C4—41no—
168C4—41no—
169C4—42no—
170C2—21no—
171C2—21no—
172C2—21no—
173C2—81no—
174C2—81no—
175C2—82no—
176C2C2×AΣL(1,𝔽16)11yescharacteristic
subgroupG/H
center, socleC2C2×AΣL(1,𝔽16)
commutatorC2×D4C22×C4
Frattini, cosocle#124C23
prime radicalC2×H(5,𝔽2)C4
LCSG ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×#124 ◁ G
LECSG ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#355C2semidirect
3. #1533C2semidirect
4. #1403C2semidirect
17. #355C22semidirect
19. #329C22semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#4552
C2×#3552
#15332
#14032